Mathematics and Social Networks

Mathematics and Social Networks
Mason A. Porter (@masonporter)
Department of Mathematics, UCLA
(2007–2016: Tutor in Applied Mathematics, Somerville College)

Outline
• Introduction and a Few Ideas
• Pictures of social networks
• Types of networks and mathematical representations
• Some old questions about social networks
• What is a complex system?
• Small worlds
• Which nodes are important?
• Multilayer Networks
• Introduction
• Which nodes are important?
• Conclusions

Introduction
• Some motivation, general ideas, and examples

• A network has nodes
(representing entities), which are
connected by edges (representing
ties between the entities). The
simplest type of network is a
graph.
• Example:
• Members of a karate club
connected to others in the club
with whom they hung out (left
figure)
• “The harmonic oscillator of
network science”
Zachary Karate Club

Types of Networks
• Binary networks: 1 if there is a connection and 0 if there isn’t
• Weighted networks: Some value if there is a connection (representing
strength of connection) and otherwise 0
• Directed networks: awkward?
• Bipartite networks: only nodes of different types are connected to each other
(e.g. an actor connected to a movie in which he/she appeared)
• Time-dependent networks: nodes and/or edges (existence and/or weights) are
time-dependent
• Multiplex networks: more than one type of edge
• Spatial networks: embedded in space
• More…

„ Representing a Network
„ Adjacency matrix A
„ This example: binary
(“unweighted”)
„ Aij = 1 if there is a connection
between nodes i and j
„ Aij = 0 if no connection
„ How do we generalize this
representation to weighted,
directed, and bipartite
examples?
Representing a Network

A Couple of Longstanding Questions in
the Study of Social Networks
• I. de Sola Pool & M. Kochen [1978–79],
“Contacts and Influence”, Social Networks,
1: 5–51 (though a preprint for 2 decades)
man through the right channels, and the more channels one has in reserve.
the better. Prominent politicians count their acquaintances by the thousands.
They run into people they know everywhere they go. The experience of
casual contact and the practice of influence are not unrelated. A common
theory of human contact nets might help clarify them both.
No such theory exists at present. Sociologists talk of social stratification;
political scientists of influence. These quantitative concepts ought to lend
themselves to a rigorous metric based upon the elementary social events of
man-to-man contact. “Stratification” expresses the probability of two people
in the same stratum meeting and the improbability of two people from dif-
ferent strata meeting. Political access may be expressed as the probability
that there exists an easy chain of contacts leading to the power holder. Yet
such measures of stratification and influence as functions of contacts do not
exist.
What is it that we should like to know about human contact nets?
-~-For any individual we should like to know how many other people he
knows, i.c. his acquaintance volume.
- For a popnfatiorl we want to know the distribution of acquaintance
volumes, the mean and the range between the extremes.
_ We want to know what kinds of people they are who have many con-
tacts and whether those people are also the influentials.
,.- We want to know how the lines of contact are stratified; what is the
structure of the network?
If we know the answers to these questions about individuals and about the
whole population, we can pose questions about the implications for paths
between pairs of individuals.
- How great is the probability that two persons chosen at random from
the population will know each other?
- How great is the chance that they will have a friend in common?
- How great is the chance that the shortest chain between them requires
two intermediaries; i.e., a friend of a friend?
very stuff of politics. Influence is in large part the ability to reach the crucial
man through the right channels, and the more channels one has in reserve.
the better. Prominent politicians count their acquaintances by the thousands.
They run into people they know everywhere they go. The experience of
casual contact and the practice of influence are not unrelated. A common
theory of human contact nets might help clarify them both.
No such theory exists at present. Sociologists talk of social stratification;
political scientists of influence. These quantitative concepts ought to lend
themselves to a rigorous metric based upon the elementary social events of
man-to-man contact. “Stratification” expresses the probability of two people
in the same stratum meeting and the improbability of two people from dif-
ferent strata meeting. Political access may be expressed as the probability
that there exists an easy chain of contacts leading to the power holder. Yet
such measures of stratification and influence as functions of contacts do not
exist.
What is it that we should like to know about human contact nets?
-~-For any individual we should like to know how many other people he
knows, i.c. his acquaintance volume.
- For a popnfatiorl we want to know the distribution of acquaintance
volumes, the mean and the range between the extremes.
_ We want to know what kinds of people they are who have many con-
tacts and whether those people are also the influentials.
,.- We want to know how the lines of contact are stratified; what is the
structure of the network?
If we know the answers to these questions about individuals and about the
whole population, we can pose questions about the implications for paths
between pairs of individuals.
- How great is the probability that two persons chosen at random from
the population will know each other?
- How great is the chance that they will have a friend in common?

Spread of “Fake News” on Social Networks

What is a Complex System?
• Complexity*:
• “The behaviour shown by Complex Systems”
• Complex System*:
• “A System Whose Behaviour Exhibits Complexity”
• I wrote a brief introduction for a general audience on Quora:
https://www.quora.com/How-do-I-explain-to-non-mathematical-
people-what-non-linear-and-complex-systems-mean
* Neil Johnson, Two’s Company,Three is Complexity, Oneworld Publications, 2007.

What is a Complex System?
My definition follows the way that the US Supreme
Court once defined pornography:
“I shall not today attempt further to define the
kinds of material I understand to be embraced within
that shorthand description [‘complex systems’]; and
perhaps I could never succeed in intelligibly doing so.
But I know it when I see it.”
(adapted from Justice Potter Stewart)

What are complex systems?
No rigorous definition; some potential features/components*:
• The system has a collection of many interacting objects or “agents”
• These objects may be affected by memory or feedback
• The system may be “open” (influenced by environment)
• The system exhibits “emergent phenomena”
• Emergent phenomena arise without a central controller
* Neil Johnson, Two’s Company,Three is Complexity, Oneworld Publications, 2007.

Two Fundamental Ideas in Complex Systems
• Emergence: Finding (possibly unexpected) order in high-
dimensional systems (large systems of interacting
components)
• We seek examples of “emergence” in the study of networks.
• Chaos: Finding (possibly unexpected) disorder in low-
dimensional systems
• E.g. Lorenz attractor

Networks are Complex Systems
• We want to somehow summarize the information in networks to learn
something about them.
• Important entities?
• Important interactions?
• Dense sets (“communities”) of entities?
• Perhaps connected sparsely to other dense sets?
• Sets of behaviorally similar entities?
• Structural bottlenecks to dynamical processes (e.g. disease or rumor spreading)
• Where should you start a rumor to maximize how far it spreads? And how does
this depend on the network structure?
• Voting dynamics on networks (“echo chambers” and “majority illusion”)
• Etc.

Watts–Strogatz “Small World” Model
• Watts–Strogatz ‘small world’ model (reviewed in MAP, Scholarpedia,
2012)
• Start with a 1D ring of nodes but connect each node to its k
nearest neighbors on each side. Then either rewire nodes with
probability p (original model) or add ‘shortcut’ with probability p
(Newman–Watts variant).
• What happens as p increases from 0?

Regime with both short mean
geodesic path length (“small world”)
and high mean clustering coefficient
A small number of shortcuts has a
small effect on local clustering, but
very quickly shortens the mean
geodesic distance from scaling linearly
to scaling logarithmically with the
number of nodes.
Note: Navigation of networks
efficiently is even more amazing
than the fact that the world is
often small.

Determining Important (“Central”) Nodes
Example: Hubs and Authorities
• J. M. Kleinberg, Journal of the ACM, Vol. 46: 604–632 (1999)
• Intuition: A Web page (node) is a good hub if it has many hyperlinks (out-edges)
to important nodes, and a node is a good authority if many important nodes have
hyperlinks to it (in-edges)
• Imagine a random walker surfing the Web. It should spend a lot of time on
important Web pages. Equilibrium populations of an ensemble of walkers satisfy
an eigenvalue problem:
• x = aAy ; y = bATx è ATAy = λy & AATx = λx, where λ = 1/(ab)
• Leading eigenvalue λ1 (strictly positive) gives strictly positive authority vector x and
hub vector y (leading eigenvectors)
• Node i has hub centrality xi and authority centrality yi

Application: Ranking Mathematics Programs
• Apply the same idea to mathematics departments based
on the flow of Ph.D. students
• S. A. Meyer, P. J. Mucha, & MAP, “Mathematical genealogy
and department prestige”, Chaos, Vol. 21: 041104 (2011)
• One-page paper in Gallery of Nonlinear Images
• Data from Mathematics Genealogy Project

Hubs and Authorities Among
US Mathematics Programs
• We consider MPG data in the US from 1973–
2010 (data from 10/09)
• Example: I earned a PhD from Cornell and
subsequently supervised students at Oxford
and UCLA.
• è Directed edge of unit weight from Cornell to
Oxford (and also from Cornell to UCLA)
• A university is a good authority if it hires
students from good hubs, and a university is
good hub if its students are hired by good
authorities.
• Caveats
• Our measurement has a time delay (only have the
Cornell è UCLA edge after I supervise a PhD
student there; normally there’s a delay)
• Eventually, there will be an edge from Oxford to
University of Vermont when Puck Rombach
graduates her first Ph.D. student. (I want
grandstudents!)
• Hubs and authorities should change in time

Geographically-Inspired Visualization
Mathematical genealogy and department
prestige
Sean A. Myers,1
Peter J. Mucha,1
and Mason A. Porter2
1
Department of Mathematics, University of North Carolina,
FIG. 1. (Color) Visualizations of a mathematics genealogy network.
CHAOS 21, 041104 (2011)
Hubs: node size
Authorities: node color

How do our rankings do?
rtment
n A. Porter2
olina,
LB, UK
2011)
(http://www.
000 scholars
elated ﬁelds.
graduation
s. The MGP
used to trace
rant, Hilbert,
We use a “geographically inspired” layout to balance node
locations and node overlap. A Kamada-Kawai visualization4
Visualizations of a mathematics genealogy network.
FIG. 2. (Color) Rankings versus authority scores.

Multilayer Networks
Review Article: M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson,Y. Moreno, &
MAP, “Multilayer Networks”, Journal of Complex Networks, 2(3): 203–271, 2014.

General Form of a Multilayer Network
• Definition of a multilayer network M
– M = (VM,EM,V,L)
• V: set of nodes (“entities”)
– As in ordinary graphs
• L: sequence of sets of possible layers
– One set for each additional “aspect” d ≥ 0 beyond an ordinary network
(examples: d = 1 in schematic on this page; d = 2 on last page)
• VM: set of tuples that represent node-layers
• EM: multilayer edge set that connects these tuples
• Note 1: allow weighted multilayer networks by
mapping edges to real numbers with w: EM èR
• Note 2: d = 0 yields the usual single-layer
(“monolayer”) networks

Supra-Adjacency Matrices
(from “flattening” tensors)
• Schematic from M. Bazzi, MAP, S. Williams, M.
McDonald, D. J. Fenn, & S. D. Howison [2016]
Multiscale Modeling and Simulation: A SIAM
Interdisciplinary Journal, 14(1): 1–41
13
Layer 1
11 21
31
Layer 2
12 22
32
Layer 3
13 23
33
!
2
6
6
6
6
6
6
6
6
6
6
6
6
4
0 1 1 ! 0 0 0 0 0
1 0 0 0 ! 0 0 0 0
1 0 0 0 0 ! 0 0 0
! 0 0 0 1 1 ! 0 0
0 ! 0 1 0 1 0 ! 0
0 0 ! 1 1 0 0 0 !
0 0 0 ! 0 0 0 1 0
0 0 0 0 ! 0 1 0 1
0 0 0 0 0 ! 0 1 0
3
7
7
7
7
7
7
7
7
7
7
7
7
5
Fig. 3.1. Example of (left) a multilayer network with unweighted intra-layer connections (solid
lines) and uniformly weighted inter-layer connections (dashed curves) and (right) its corresponding
adjacency matrix. (The adjacency matrix that corresponds to a multilayer network is sometimes
called a “supra-adjacency matrix” in the network-science literature [39].)
or an adjacency matrix to represent a multilayer network.) The generalization in [49]
consists of applying the function in (2.16) to the N|T |-node multilayer network:
N|T | ✓ ◆

Example: Multiplex Network
(e.g. multirelational social network)
• Monster movement in the
game “Munchkin Quest”

My Ego-Centric Multiplex Network
(edge-colored multigraph)

Example: Interconnected Network
(e.g. UK infrastucture)
(CourtesyofScottThacker,ITRC,UniversityofOxford)

Time-dependent centrality from multilayer
representation of time-dependent network
• Math department rankings change in time, so we need centrality
measures that change in time
• E.g. via a multilayer representation of a temporal network
• Multilayer network with adjacency tensor elements Aijt
• Directed intralayer edge from university i to university j at time t for a
specific person’s PhD granted at time t for a person who later advised a
student at i (multi-edges give weights)
• E.g. I would yield an OxfordèCornell edge and a UCLAèCornell edge for t =
2002
• Though neither of these is actually in the analyzed data set
• Use a multilayer network with “diagonal” and ordinal interlayer coupling
• 231 US universities, T = 65 time layers (1946–2010)

Ranking Math Programs: Best Authorities
• Note: We construct a “supra-centrality matrix” and do a singular perturbation
expansion (and examine the coefficients of the expansion).
18 S. A. MYERS et al.
Table 4.1
Top centralities and ﬁrst-order movers for universities in the MGP [4].
Top Time-Averaged Centralities Top First-Order Mover Scores
Rank University ↵i
1 MIT 0.6685
2 Berkeley 0.2722
3 Stanford 0.2295
4 Princeton 0.1803
5 Illinois 0.1645
6 Cornell 0.1642
7 Harvard 0.1628
8 UW 0.1590
9 Michigan 0.1521
10 UCLA 0.1456
Rank University mi
1 MIT 688.62
2 Berkeley 299.07
3 Princeton 248.72
4 Stanford 241.71
5 Georgia Tech 189.34
6 Maryland 186.65
7 Harvard 185.34
8 CUNY 182.59
9 Cornell 180.50
10 Yale 159.11
map: do we want to indicate what any of those other papers do with
the MGP data?drt: my vote is no. keep things as brief as possible.
We extend our previous consideration of this data [71] by keeping the year that
each faculty member graduated with his/her Ph.D. degree. We thus construct a
multilayer network of the MGP Ph.D. exchange using elements Aijt that indicate a
directed edge from university i to university j at time t to represent a doctoral degree

Conclusions
• Mathematical ideas have long been used in the
study of social networks.
• ”Network science” is a vibrant and ever-growing
area of mathematics.
• Note: major connections to data science, graph theory,
probability, statistics, dynamical systems, statistical
mechanics, optimization, computer science, and more
• The study of “multilayer networks”, currently the
most prominent area of network science, allows
one to examine (and integrate) heterogeneous
types of information.

Mathematics and Social Networks

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